University:
Carleton College
Course:
MATH 101 | Calculus with Problem Solving
Academic year:

2024
Views:

466

Pages:

3
Author:

Clint W.

8.6 Exponential Decay Functions Notes How is exponential decay different from exponential growth? Exponential Decay Function: y = a * b^x growth decay b > 1 0 < b < 1 Example 1: Write a rule for the function. Decide if the function represents growth or decay. 1. | x | -2 | -1 | 0 | 1 | 2 | |---|----|----|---|---|---| | y | 16 | 4 | 1 | 1/4 | 1/16 | y = 1 * (1/4)^x 2. | x | -2 | -1 | 0 | 1 | 2 | |---|---|----|---|---|---| | y | 16 | 4 | 1 | 1/4 | 1/16 | y = 1 * (1/4)^x Example 2: Graph the exponential function y = (1/3)^x. Then identify its domain and range. [Image of the graph of y = (1/3)^x] Domain: all real numbers Range: y > 0 Example 4: Graph the functions and compare them. y = (1/2)^x y = 3 * (1/2)^x y = -1 * (1/2)^x [Table of values for the three functions] [Graph of the three functions] Summary: getting bigger/getting smaller, or b > 1 is decay, b < 1 is growth Example 5: Write an exponential function of the form y = ab^x whose graph passes through the points (0, 4) and (2, 1/4). [Image of a table with x and y values, and calculations] y = a * b^x y = 4 * (1/4)^x 4 * b^2 = 1/4 b^2 = 1/16 b = 1/4 Exponential Growth Model Growth rate (decimal) y = a(1+r)^t Amount Decay y = a(1-r)^t Example 5: Identify the growth rate, growth factor, and initial amount. a. y = 350(1 + 0.3)^t Growth rate: 0.3 Growth factor: 1.3 Initial amount: 350 b. y = 56(1.04)^t Growth rate: 0.04 Growth factor: 1.04 Initial amount: 56 Example 6: You put $250 in a savings account that earns 4% annual interest compounded yearly. You do not make any deposits or withdrawals. How much will your investment be worth in 10 years? y = 250(1 + 0.04)^10 ≈ $370.06 Example 7: The owner of an original copy of a 1938 comic book sold it at an auction in 2005. The owner bought the comic book for $55 in 1980. The value of the comic book increased at a rate of 2.8% per year. a. Write a function that models the value of comic book over time. y = 55(1 + 0.028)^t b. What was the approximate value of the comic book at the time of the auction in 2005? Round your answer to the nearest dollar. y = 55(1.028)^25 ≈ $109.70 Ex) Buy a car for $13,000. Depreciates at 15% per year. How much worth after 4 years? y = 13,000(1 - 0.15)^4 = 13,000(0.85)^4 y ≈ $6,786.08